Conductors and newforms for \(SL(2)\)
Let \(F\) be a non-Archimedean local field of odd residue characteristic. The theory of newforms of a representation, that is vectors fixed by congruence subgroups, was developed for \(GL_2(F)\) by \textit{W. Casselman} [Math. Ann. 201, 301--314 (1973; Zbl 0239.10015)], and for \(GL_n(F)\) by \textit{J. Hervé, I. I. Piatetski-Shapiro} and \textit{J. Shalika} [Math. Ann. 256, 199--214 (1981; Zbl 0443.22013)]. Newforms are important for the theory of automorphic forms; see the survey by \textit{R. Schmidt} [J. Ramanujan Math. Soc. 17, No.~2, 115--147 (2002; Zbl 0997.11040)]. The authors develop a theory of newforms for \(SL_2(F)\). They introduce and study conductors for the irreducible representations; the connection with conductors for \(GL_2\) is found. The dimension of the space of newforms is shown to be one or two. It is proved that the appropriate Whittaker functionals are nonzero on spaces of newforms.
scientific article; zbMATH DE number 5366235